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Title: Cyclotomic matrices and graphs
Author: Taylor, Graeme
ISNI:       0000 0004 2727 2495
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2010
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We generalise the study of cyclotomic matrices - those with all eigenvalues in the interval [-2; 2] - from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers of imaginary quadratic fields. As in the rational integer case, a corresponding graph-like structure is defined. We introduce the notion of `4-cyclotomic' matrices and graphs, prove that they are necessarily maximal cyclotomic, and classify all such objects up to equivalence. Six rings OQ( p d) for d = -1;-2;-3;-7;-11;-15 give rise to examples not found in the rational-integer case; in four (d = -1;-2;-3;-7) we recover infinite families as well as sporadic cases. For d = -15;-11;-7;-2, we demonstrate that a maximal cyclotomic graph is necessarily 4- cyclotomic and thus the presented classification determines all cyclotomic matrices/graphs for those fields. For the same values of d we then identify the minimal noncyclotomic graphs and determine their Mahler measures; no such graph has Mahler measure less than 1.35 unless it admits a rational-integer representative.
Supervisor: Smyth, Chris. Sponsor: Engineering and Physical Sciences Research Council (EPSRC) ; Maxwell Institute for Mathematical Sciences ; University of Edinburgh ; Edinburgh Compute and Data Facility (ECDF)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: cyclotomic matrices ; rational integer ; infinite families